Altland and Zirnbauer Reply:

Altland, Alexander; Zirnbauer, Martin R.

doi:10.1103/physrevlett.80.641

Cited by 351 publications

(759 citation statements)

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“…This result relates the orthogonal pre-kernel to the unitary kernel K 2a 2n (x, y) which has been studied elaborately in the literature [43][44][45]. The relation (22) is exact for a Gaussian potential in which case it coincides with the result obtained in [28,47,46,10,31].…”

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confidence: 65%

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Universality in Chiral Random Matrix Theory atβ=1andβ=4

Sener

Verbaarschot

1998

Phys. Rev. Lett.

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In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real (β = 1) and quaternion real (β = 4) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles (β = 2). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Brézin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for β = 2 as shown by Akemann, Damgaard, Magnea and Nishigaki.PACS numbers: 64.60. Cn, 05.45.+b, 11.30.Rd, 12.38.Aw Since its introduction in nuclear physics [1], Random Matrix Theory (RMT) has been applied successfully to many different branches of physics ranging from atomic physics to quantum gravity (for a recent comprehensive review we refer to [2]). One important common ingredient is that eigenvalue correlations appear to be insensitive to the details of the underlying Hamiltonian. The success of RMT is based on this type of universality, and it is no surprise that it has received a great deal of attention in recent literature [3-10] [11-20] [21-27]. What has been shown is that spectral correlators on the scale of the average eigenvalue spacing are insensitive to the details of the probability distribution of the matrix elements. Because of its mathematical simplicity most studies were performed for complex (β = 2) Hermitean RMT's. However, in the case of the classical RMT's it was shown that universality extends to real (β = 1) and quaternion real (β = 4) matrix ensembles [4,6,7]. This suggests that relations between correlation functions for different values of β which can be derived for a Gaussian probability distribution [28,29] might be valid for a wide class of probability distributions. The main goal of this paper is to establish such general relations. As a consequence, universality for the much simpler complex ensembles implies universality for the real and quaternion real ensembles.In this letter we address the question of microscopic universality for the chiral ensembles. These ensembles are relevant for the description of spectral correlations of the QCD Dirac operator. They also appear in theory of universal conductance fluctuations in mesoscopic systems [30,31]. In particular, they can be applied to spectral correlations near λ = 0. According to the Banks-Casher formula [32], this part of the spectrum is directly related to the order parameter Σ of the chiral phase transition (Σ = lim πρ(0)/V , where V is the volume of space time and ρ(λ) = k δ(λ−λ k )). It is therefore natural to introduce the microscopic limit where the variable u = λV Σ is kept fixed for V → ∞. For example, the microscopic spectral density is defined by [33] ρ S (u) = limwhere the average is over the distribution of the matrix elements of the D...

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confidence: 65%

“…These ensembles are relevant for the description of spectral correlations of the QCD Dirac operator. They also appear in theory of universal conductance fluctuations in mesoscopic systems [30,31]. In particular, they can be applied to spectral correlations near λ = 0.…”

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confidence: 99%

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Sener

Verbaarschot

1998

Phys. Rev. Lett.

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“…For this, the dictionary in [50] is useful. It would be interesting to compare this classification with the classification based on sigma models [51]. The latter classification is based on discrete symmetries such as time-reversal, so it is not as strong as our classification of the actual critical points.…”

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confidence: 99%

Strong-coupling fixed points of current interactions and disordered fermions in two dimensions

LeClair

2001

Phys. Rev. B

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The all-orders βeta function is used to study disordered Dirac fermions in 2D. The generic strong coupling fixed 'points' of anisotropic current-current interactions at large distances are actually isotropic manifolds corresponding to subalgebras of the maximal current algebra at short distances. We argue that IR fixed point theories are generally current algebra cosets. We illustrate this with the simple example of anisotropic su(2), which is the physics of Kosterlitz-Thouless transitions. We propose a phase diagram for the ChalkerCoddington network model which is in the universality class of the integer Quantum Hall transition. One phase is in the universality class of dense polymers.Typeset using REVT E X * Laboratoire associé No. 280 au CNRS 1

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“…In this way the bulk-surface connection is seen to be a manifestation of the Callan-Harvey "anomaly inflow" mechanism [3]. Most of the Altland-Zirnbauer classes [4,5] of topological insulators possess conserved U(1) charge or SU(2) spin currents, and the necessity of their protected surface modes can be understood via ordinary gauge-field anomalies. An important exception is the class DIII, which includes superconductors with spin-orbit interactions, and superfluid 3 He-B.…”

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Gravitational anomalies and thermal Hall effect in topological insulators

2012

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It has been suggested that, after being gapped by small symmetry breaking field, the Majorana quasiparticles localized on the surface of a class DIII topological insulator will exhibit a thermal Hall effect that arises from a gravitational Chern-Simons term. We critically examine this idea, and argue that the thermo-gravitational Hall effect is more complicated than its familiar analogue. A conventional Hall current is generated by a uniform electric field, but computing the flux from the gravitational Chern-Simons functional shows that gravitational field gradients -i.e. tidal forcesare needed to induce a energy-momentum flow. We relate the resulting surface energy-momentum flux to a domain-wall gravitational anomaly via the Callan-Harvey inflow mechanism. We stress that the gauge invariance of the combined bulk-plus-boundary theory ensures that the current in the domain wall always experiences a "covariant" rather than "consistent" anomaly. We use this observation to confirm that the tidally induced energy-momentum current exactly accounts for the covariant gravitational anomaly in (1 + 1) dimensional domain-wall fermions. The same anomaly arises whether we write the Chern-Simons functional in terms of the Christoffel symbol or in terms of the the spin connection.

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Altland and Zirnbauer Reply:

Cited by 351 publications

References 2 publications

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Universality in Chiral Random Matrix Theory atβ=1andβ=4

Strong-coupling fixed points of current interactions and disordered fermions in two dimensions

Gravitational anomalies and thermal Hall effect in topological insulators

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